### A dichotomy theorem for mono-unary algebras

We study the isomorphism relation of invariant Borel classes of countable mono-unary algebras and prove a strong dichotomy theorem.

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We study the isomorphism relation of invariant Borel classes of countable mono-unary algebras and prove a strong dichotomy theorem.

A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $(A,f)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form $\mathrm{M}od({f}^{n}\left(x\right)=x)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in $\mathrm{H}SC\left(\mathrm{M}od({f}^{mn}\left(x\right)=x)\right)$ for every $m>1$ where $\mathrm{C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of...

Let $T$ be an infinite locally finite tree. We say that $T$ has exactly one end, if in $T$ any two one-way infinite paths have a common rest (infinite subpath). The paper describes the structure of such trees and tries to formalize it by algebraic means, namely by means of acyclic monounary algebras or tree semilattices. In these algebraic structures the homomorpisms and direct products are considered and investigated with the aim of showing, whether they give algebras with the required properties. At...

A tolerance on an algebra is defined similarly to a congruence, only the requirement of transitivity is omitted. The paper studies a special type of tolerance, namely atomary tolerances. They exist on every finite algebra.

In this paper we find cardinalities of lattices of topologies of uncountable unars and show that the lattice of topologies of a unar cannor be countably infinite. It is proved that under some finiteness conditions the lattice of topologies of a unar is finite. Furthermore, the relations between the lattice of topologies of an arbitrary unar and its congruence lattice are established.

For an uncountable monounary algebra $(A,f)$ with cardinality $\kappa $ it is proved that $(A,f)$ has exactly ${2}^{\kappa}$ retracts. The case when $(A,f)$ is countable is also dealt with.

An M-Set is a unary algebra $\langle X,M\rangle $ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M\rangle $ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M\rangle $ if the congruence lattice of $\langle X,M\rangle $ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\Pi \left(L\right)$ is introduced here. $\Pi \left(L\right)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\Pi \left(L\right)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi $-product lattice. A $\Pi $-product...